For questions involving random variables uniformly distributed on a subset of a measure space. To be used with [probability] or [probability-theory] tag.

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39 views

Moment of uniform distribution

Suppose that $U$ is a random variable from a uniform distribution on $[a, b]$. Then, we can obtain the moment generating function of $U$, and by using that, we can get the $n$th order moment of $U$ ...
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2answers
372 views

Uniform Distribution in [0,1] where P[x1+x2<=x3]

Consider the following question : X1, X2, X3 are 3 independent random variables having uniform distribution between [0,1] then P[x1+x2<=x3] to the greatest value is ? Now this is not a homework. ...
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1answer
34 views

Proving a process generates a uniform distribution

I have a process that generates a series of real numbers. Specifically, starting from a given arbitary value (Xi-1), the process will generate a new number Xi following the formula: Xi = Xi-1 + ...
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2answers
129 views

A Brownian motion $B$ that is discontinuous at an independent, uniformly distributed random variable $U(0,1)$

Suppose that $\left\{B\left(t\right): t \geq 0\right\}$ is a Brownian motion and $U$ is an independent random variable, which is uniformly distributed on $\left[0,1\right]$. Then the process ...
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1answer
36 views

If $X$ ~ $U[0, 4]$ and $Y$~$[0, 7]$ find the probability X value is greater than Y value

Suppose $X$ and $Y$ are continuous uniform random variables. If $X$ ~ $U[0, 4]$ and $Y$~$[0, 7]$ find the probability that a random $X$ value is greater than a random $Y$ value.
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51 views

Choosing points uniformly on a sphere surface

I need a set, $A$, of points on a sphere surface, $S$. $A$ must satisfy: 1. The mean is the exact center of the sphere. 2. $\forall p_1,p_2\in S:\sum _{i\in A} \text{od}\left(i,p_1\right)=\sum _{i\in ...
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124 views

Product of standard normal and uniform random variable

I'm trying to find the PDF of the product of two random variables by first finding the CDF. I don't know where I'm going wrong. Let $X\sim N(0,1)$ and $Y\sim Uniform\{-1,1\}$ and let $Z = XY$, then: ...
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1answer
112 views

Sum of three numbers from unformly distributed set equals to zero

I'm reading Sedgewick's "Algorithms" and completely stuck at one exercise. It is formulated like that: Develop an appropriate mathematical model describing the number of triples of N random int ...
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1answer
28 views

CDF on Standard uniform gives the same distribution

Assume that $X$ has a continuous and strictly increasing CDF $F_X$. Define $Y = F_X^{-1}(U)$ where $U$ is standard Uniform. How dow I show that $X$ and $Y$ have the same distribution?
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25 views

Computing an expectation with a uniform probability distribution.

Suppose $F$ is a cumulative density function of a uniform distribution between $a=0$ and $b=B$ and $c$ is a positive real number. I need to evaluate the integral $$\int_c^Bq\;\mathrm dF$$ where the ...
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1answer
40 views

Uniformed Distribution - Recap

I have divide the interval $[0,1]$ into $k$ equal sub-intervals, which I call classes, and generated $n$ observations from a uniform distribution. The number $X_{1}$ of the $n$ observations that fall ...
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63 views

What is the conditional distribution of this random vector?

Let us have random vectors $X_1, \dots, X_N$ which are identically independently uniformly distributed in the $n$-dimensional unit hyperbox $[0; 1]^n$. Let $c = (0.5, \dots, 0.5)$ be the center of ...
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1answer
40 views

Method of moments on uniform distributions

I need help on how to find the estimates $a$ and $b$ in the uniform distribution $\mathcal U[a,b]$ using the method of moments. This is where I am at: I have found $U_1=\overline X$ and ...
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2answers
57 views

Distribute a population size based on fractions using random number generator drand48()

I have a population size of say 5000 people. Every person belongs to either A, B, C or D category. I want to split the population as per a given fraction provided by user. for example, 99% of A, 0.4% ...
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2answers
171 views

$X$ and $Y$ are uniformly ditributed on $(0,1)$. distribution of $\max(X,Y)/\min(X,Y)$

Suppose that $X$ and $Y$ are chosen randomly and independently according to the uniform distribution from $(0,1)$. Define $$ Z=\frac{\max(X,Y)}{\min(X,Y)}.$$ Compute the probability distribution ...
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2answers
289 views

Transformation of a uniform distribution in order to get a random variable distributed like Y.

$f(y)=\begin{cases} \frac{b}{y^2}, & y\ge b,\\ 0, & \mbox{elsewhere}\end{cases}$. is a bona fide probability density function for a random variable, $Y$. Assuming $b$ is a known constant and ...
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2answers
76 views

Calculate expectation and variance

Let $(X_n)$ be a sequence of independent RVs which are uniformly distributed on $[0,1]$ interval. For $0<x\le 1$ we define $$N(x):=\inf\{n:X_1+\dots+X_n\ge x\}.$$ Show that $$\mathbb{P}(N(x)\ge ...
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1answer
68 views

What's the distribution of the exponential of uniformly distributed variable?

I want to know the distribution of $z = \exp(j\varphi)$, with $\varphi \sim \mathcal{U}[-\pi;+\pi]$. From the book "Probability, Random Variables and Stochastic Processes" by Papoulis and Pillai I ...
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0answers
122 views

Expected Value - Uniform distribution over infinite interval

Question: The probability that an error is introduced into a packet is $\alpha$. Messages, consisting of one or more packets, are received at a node. Given that a message has been received free of ...
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2answers
75 views

Probability with multiple uniform distributions

Question: Two sources output a number at equal rates. The output from source A is uniformly distributed between 100 and 199, and the output from source B is uniformly distributed between 50 and 249. ...
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2answers
84 views

Calc expected value of 5 random number with uniform distribution

Assume we have a random numbers $\sim U(0,100)$. Then the expected value of that number will be: $\int_{0}^{100} \frac{x}{100}$ = 50.5 Now assume we have 5 random numbers $\sim U(0,100)$. How can I ...
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3answers
173 views

uniform moment generating function at t=0

I have calculated the moment generating function for the uniform distribution as Mx(t) = ((e^(tb)-e^(ta))/t(b-a) However I know Mx(0)=1 but I can't get my head around how this is possible as if t=0, ...
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1answer
886 views

Probability the three points on a circle will be on the same semi-circle

Three points are chosen at random on a circle. What is the probability that they are on the same semi circle? If I have two portions $x$ and $y$, then $x+y= \pi r$...if the projected angles are $c_1$ ...
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0answers
47 views

Prove that $E(X_1 \dots X_n)^\frac{1}{n} \leq (EX_1 \dots EX_n)^\frac{1}{n}$, with $X_i$ uniform distribution

Let $X_1, \dots , X_n$ i.i.d. uniformly distributed random variables with $f(x) = 1_{(0,1)}(x)$, $x \in \mathbb{R}$. Let $\Pi_n = (X_1 \dots X_n)^\frac{1}{n}$ and $M_n = \max \{ X_1, \dots , X_n ...
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1answer
45 views

Finding the joint density of $Z=X+Y$ where $X\in U(0,1), Y\in U(0,\alpha)$

I'm trying to find the joint density of $Z=X+Y$ where $X\in U(0,1), Y\in U(0,\alpha)$ Here $U$ is the uniform distribution. The method I use i to introduce an auxilary variable $W=X$ and then use ...
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1answer
237 views

Distribution function of the sum of poisson and uniform random variable.

Merry Christmas to everybody. I am working on the following problem. Let $X$ and $Y$ be independent Poisson($\lambda$), respectively Uniform$(0,1)$ random variables. Find the distribution function of ...
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2answers
77 views

How to show $\max\{Y_{1},Y_{2},\cdots,Y_{n}\}$ converges in probability to $\theta$ as $n \to \infty$.

Let $Y_{1},Y_{2},\ldots,Y_{n} $ be independent random variables , each uniformly distributed over the interval $(0,\theta)$. Show that $\max\{Y_{1},Y_{2},\ldots,Y_{n}\}$ converges in probability to ...
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1answer
34 views

Simple probability problem.

$X$ ~ $U(0,1)$ and $Y$~U$(0,1)$ are two indenpendent variables. Get Pr ( Y > X). NOW what i don't understant in this problem is how you set the limits of integration. I heard that you must set ...
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1answer
263 views

Conditional uniform distribution

I had this question in a quiz, and now that I am reviewing it, I am not sure if why my TA gave me the marks because I am pretty sure I am wrong. Let the r.v. $Y$ follow uniform distribution $U(1,2)$ ...
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1answer
88 views

Finding density function for uniform distribution

Can anyone help me set this up correctly, please: John is going to eat at at McDonald's. The time of his arrival is uniformly distributed between 6PM and 7PM and it takes him 15 minutes to eat. Mary ...
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1answer
332 views

Show that the nth order statistic is a consistent estimator of a uniform parameter

We assume that our observations come from a uniform $(0,\theta)$ distribution. Can you please check my work on the following? We can derive the distribution function of the maximum of the sample, ...
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1answer
465 views

Question about the Irwin-Hall Distribution (Uniform Sum Distribution)

So I have been reading about the Irwin-Hall distribution online, it is a sum of uniform distributions on $[0,1]$, and it seems very interesting: ...
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1answer
226 views

Density of uniformly chosen random point inside triangle

Imagine the triangle inside of the points $(0,0), (0,1)$ and $(1,0).$ Let $(X,Y)$ be a uniformly chosen random point from the triangle. Then find the joint density of $X$ and $Y$. The answer is ...
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1answer
41 views

Weak convergence and limiting distribution

I have $X_{i} \sim \operatorname{Unif}\left(0,1\right)$ iid random variables and have to show that $$ \frac{4\sum_{i=1}^n iX_{i} - n^2}{n^{3/2}}$$ converges weakly and compute its limit. How can I do ...
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2answers
76 views

Find the probability of $ x_2/x_3 \leq a $ where $x_2,x_3$ are uniform i.i.d.

Let $x_1,x_2,...,x_n $ be independent and identically distributed, uniformly on $(0,1)$. How to find $P(x_2/x_3 \leq a)$?
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1answer
253 views

Sufficient statistic for uniform distribution

Given random sample $\left\{ { X }_{ 1 },{ X }_{ 2 },...,{ X }_{ n } \right\} $ from $ U(0,\theta)$. Let ${Y}_{i}$ be the order statistics. Then the sufficient statistic for $\theta$ is ${ Y }_{ n ...
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1answer
93 views

Bivariate and Multivariate Probability Distributions

For my homework for Bivariate and Multivariate Probability Distributions section, I encounter the terms joint density, joint distributed random variable, joint probability, uniform distribution, when ...
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2answers
374 views

Uniform distribution over the unit circle

Suppose that $U$ and $V$ are two independent uniform $(-1,1)$ random variables. Any hints on how I can show that their conditional distribution, given $U^2 +V^2<1$ is given by the uniform ...
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1answer
75 views

$(\log n)_{n \in \mathbf{N}}$ not uniformly distributed mod 1.

Let $(x_n)_{n \in \mathbf{N}}$ be a sequence of real numbers, we say that $(x_n)$ is uniformly distributed mod 1 (u.d. mod 1) if $$\lim_{N \to \infty} \frac{|\{1\leq n \leq N : (x_n -\lfloor x_n ...
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0answers
72 views

Interval of non-uniformly distributed set of numbers adjusted that it properly excludes extremes

Let's say I have an interval of numbers from 1 to 9 with the following frequency of distribution: numbers 1, 2 and 3 about 20 occurrences number 6 has 2 occurrences and number 9 has only ...
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1answer
36 views

Probability that at least one event (out of two uniform RV) happens before two other random events

I recently faced a probability problem that is puzzling me. I would like to ask you if you could help me. I have two random variables X1 and X2 i.i.d with uniform distribution U[64,96] and other two ...
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1answer
93 views

How to find the pdf of the minimum of absolute differences of Uniform distributions.

Let $X_1$,$X_2$ and $X_3$ are independent random variables that are uniformly distributed over $(0;b), b>0$. What is the probability density function of z=min($Y_1$,$Y_2)$, where $Y_1=|X_1-X_2|$ ...
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1answer
42 views

Looking for a simple bivarate uniform distribution with non-zero covariance matrix

Obviously there are many forms this can take, I'm looking for on that gives an non-zero (off diagonal elements) covariance matrix. Does anyone know of one?
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1answer
31 views

Uniformly Distributed ingredients

Suppose we need to make a dish that has three ingredients A, B and C. All are distributed uniformly between [0, 2], [0, 2], [0, 1] respectively. To create the dish, we need 1/4 of A, 1/4 of B and 1/8 ...
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1answer
28 views

Finding the joint density of two random variables

Suppose (X,Y) is uniformly distributed over the region { (x, y) : 0 < x < y < 1 }. Find the joint density of (X, Y). I started out by drawing the unit square and filling in the area where 0 ...
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0answers
79 views

How to construct a uniform joint distribution

I have a question that is critical to my work, but I am not sure if it is any possible. Assume that you have two uniform random variables X and Y. The product distribution of Z=XY is not a uniform. ...
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1answer
68 views

Finding probability of uniform random variable given a condition with another random variable

Suppose X and Y are independent and uniformly distributed on the unit interval (0,1). Find: $$P[Y>\frac{1}{2}\,|\,Y>1-2X]$$ How I approached it was to find the area where $Y > 1 - 2X$, and ...
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1answer
151 views

Probability of maximum of 2 uniform random variables

The random variables X and Y are independent, each with the uniform distribution on [−1, 1]. Find: $$P[max (X,Y) >0.5]$$ Apparently there is an easy approach without integration, but I am having ...
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37 views

choose a list of words such that have equal letter frequency

I have a big list meaning full Words. surely letter frequency of this word list is different for each letter. Now my problem is to find a way to randomly select words from this word list to a new ...
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2answers
247 views

Expected value of the function of a uniformly distributed random variable

Let X by a uniformly distributed random variable on the interval [0,1]. Find $E[e^Y]$ I am trying to make use of the formula $$E[g(X)] = \int_{-\infty}^{\infty}g(x)xdx$$ so then $$E[e^X] = \int ...